Social choice of convex risk measures through Arrovian aggregation of variational preferences

Institute of Mathematical Economics Working Papers

May 2010

432

Social choice of convex risk measures

through Arrovian aggregartion of variational preferences

Frederik Herzberg

IMW·Bielefeld University Postfach 100131 33501 Bielefeld·Germany

Social choice of convex risk measures through Arrovian aggregation of variational preferences ^∗

Frederik Herzberg

Abstract

This paper studies collective decision making with regard to convex risk measures: It addresses the question whether there exist non- dictatorial aggregation functions of convex risk measures satisfying Arrow-type rationality axioms (weak universality, systematicity, Pareto principle). Herein, convex risk measures are identied with variational preferences on account of the MaccheroniMarinacciRustichini (2006) axiomatisation of variational preference relations and the Föllmer Schied (2002, 2004) representation theorem for concave monetary utility functionals.

We prove a variational analogue of Arrow's impossibility theorem for nite electorates. For innite electorates, the possibility of rational aggregation depends on a uniform continuity condition for the variational preference proles; we prove variational analogues of both Campbell's impossibility theorem and Fishburn's possibility theorem. The proof methodology is based on a model-theoretic approach to aggregation theory inspired by LauwersVan Liedekerke (1995).

An appendix applies the DietrichList (2010) analysis of majority voting to the problem of variational preference aggregation.

Key words: Arrow-type preference aggregation; judgment aggregation;

abstract aggregation theory; variational preferences; multiple priors preferences; convex risk measure; model theory; rst-order predicate logic;

ultralter; ultraproduct

2010 Mathematics Subject Classication: 91B14, 91B16, 03C20, 03C98 Journal of Economic Literature classication: D71, G11

∗This work has been partially supported by a German Research Foundation (DFG) grant while the author visited the Mathematics Department of Princeton University. A talk based on this paper was presented at the 10th Society for the Advancement of Economic Theory (SAET) Conference on Current Trends in Economics in Singapore, August 2010. I would like to thank Daniel Eckert, Edward Nelson, Konrad Podczeck and Frank Riedel for discussions and comments.

†Institut für Mathematische Wirtschaftsforschung, Universität Bielefeld, Universitätsstraÿe 25, D-33615 Bielefeld, Germany. fherzberg@uni-bielefeld.de

‡Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, New Jersey 08544-1000, United States of America. fherzberg@math.princeton.edu

1 Introduction

Convex risk measures can be represented as negated maxmin expected utility functions with additive convex lower-semicontinuous penalty (Föllmer and Schied [11, 12]), which in turn are in a one-to-one correspondence with the set of so-called variational preference relations (Maccheroni, Marinacci and Rustichini [23]). Given such an individual decision-theoretic foundation for convex risk measures, it is only natural to study the aggregation problem for convex risk measures as an aggregation problem for variational preference relations.

Whilst classical preference aggregation theory does not provide suitable methods to study the aggregation of variational preferences, the scope of aggregation theory has developed considerably during the past decade: It now encompasses aggregation problems of very general form, including even the aggregation of logical propositions.

One of the most recent developments among these generalisations of classical (Arrovian) preference aggregation theory concerns the aggregation of relational structures (model aggregation). This approach can best be seen as a continuation of Lauwers and Van Liedekerke's far-sighted paper [20] and was elaborated systematically recently by Herzberg and Eckert [15, 16].¹

It is a rather natural methodological choice to employ model aggregation theory in our analysis of variational preference aggregation, on account of the intrinsic emphasis which model aggregation lays on semantics (in comparison with most of the judgment aggregation literature) and also because of its historical roots in preference aggregation theory through the work of Lauwers and Van Liedekerke [20]. (Other general approaches to aggregation theory can be found in the literature on judgment aggregation, including the abstract aggregation theories of Nehring and Puppe [26] and of Dokow and Holzman [9]

and, in particular, the rich body of work by List and Pettit [21], Dietrich and List [5, 6, 7], and Dietrich and Mongin [8].)

This methodology enables us to prove variational analogues of three of the most important (im)possibility theorems of social choice theory: those of Arrow, Fishburn, and Campbell. Moreover, it may well be possible to apply the same proof methodology to obtain similar results for multiple-priors preferences (which can be represented by coherent risk measures) and perhaps ultimately even for dynamic variational or multiple-priors preferences.

The paper is structured as follows: Section 2 reviews the axioms and the representation theorem of variational preferences and relates them to convex risk measures. Section 3 proposes a formal framework for an Arrovian aggregation theory of variational preferences, within which Section 4 formulates the main (im)possibility results of this paper. Section 5 then describes briey the ideas behind the proof methodology (model aggregation theory), while Section 6 discusses possible extensions and future research.

In an appendix, we also apply Dietrich and List's [7] account of majority voting to the problem of variational preference aggregation. The fruit is a possibility theorem, but at the cost of considerable and at least at rst sight rather unnatural restrictions on the domain of the variational preference aggregator.

1For another recent application of that approach to the problem of representative-agent microfoundations for certain parametrised aggregator domains see Herzberg [14].

2 Variational preferences and convex risk measures

Consider a nite set S, called the set of states of the world, letX be a convex subset of a vector space Y with more than one element, called the set of consequences, letFbe the set of all functions fromStoX. Then,Fis a convex subset of the vector spaceY^S. LetFcbe the set of all constant functions fromS toX. Every elementx∈X can be identied with the constant functions7→x in Fand thus with an element ofF_c.

Let us now introduce axioms for a binary relation%with symmetric part∼ (i.e. f ∼g if and only if f %g andg%f) and asymmetric part Â(i.e. f Âg if and only iff %gbutg6∼f); our formulation of the axioms is borrowed from Maccheroni, Marinacci and Rustichini [23, p. 1453].

Denition 1. A binary relation%onFwith symmetric part∼and asymmetric partÂis a variational preference ordering or convex risk-preference ordering if and only if it satises all of the following axioms:

(A1) Weak order properties. For all f, g ∈ F, either f % g or g % f (completeness); for all f, g, h ∈ F, if f % g and g % h, then f % h (transitivity).

(A2) Weak certainty independence. For all f, g∈F, x, y ∈Fc and α∈(0,1), if

αf+ (1−α)x%αg+ (1−α)x, then

αf+ (1−α)y%αg+ (1−α)y.

(A3) Continuity. For allf, g, h∈F, the sets

{β∈[0,1] : βf+ (1−β)g%h}

and

{β∈[0,1] : h%βf+ (1−β)g}

are closed.

(A4) Monotonicity. For allf, g∈F, iff(s)%g(s)for all s∈S, thenf %g.

(A5) Uncertainty aversion. For all f, g ∈ F and α ∈ (0,1), if f ∼ g, then αf+ (1−α)g%f.

(A6) Non-degeneracy. There exist f, g∈F such thatf Âg.

Remark 2. Let % be a binary relation on F with symmetric part ∼ and asymmetric part Â.

1. If%satises completeness (A1a), then f %g⇔f 6≺g for allf, g∈F.

2. If%satises completeness (A1a), then %satises continuity (A3) if and only if for all f, g, h∈ F and all β ∈ [0,1], there exist α, γ ∈[0,1] such that

• (α, γ)⊆ {δ∈[0,1] : δf+ (1−δ)g%h} if βf+ (1−β)g%h, and

• (α, γ)⊆ {δ∈[0,1] : h%δf+ (1−δ)g} if h%βf+ (1−β)g, while either

• 0≤α < β < γ≤1 or

• 0 =α=β < γ≤1 or

• 0≤α < β=γ= 1.

The identication of variational preference relations with convex risk- preference orderings can be justied as follows: On the one hand, Maccheroni, Marinacci and Rustichini [23, pp. 1453, 1456] have extended previous work by Gilboa and Schmeidler [13] and established that a relation % satisfying axioms (A1-A6) allows for a representation in terms of a maxmin expected utility function with additive convex lower-semicontinuous penalty: A binary relation%onF is a variational preference relation if and only if there exists a nonzero linear functionu:X→Rand a convex lower-semicontinuous function c : ∆ → [0,+∞] (∆ being the set of all probability measures on S) whose inmum is>−∞such that for anyf, g∈F, one has

f %g⇔min

p∈∆

µZ

u◦f dp+c(p)

¶

≥min

p∈∆

µZ

u◦gdp+c(p)

¶ .

On the other hand, Föllmer and Schied [11, 12] have demonstrated that convex risk measures can be represented as negated maxmin expected utility functions with additive convex lower-semicontinuous penalty and real consequences (i.e.

X ⊆R). Therefore, variational preference relations are the ordinal equivalents of convex risk measures.

In our investigation of aggregation of variational preference orderings (i.e.

convex risk-preference orderings), it will be helpful to have a more quantitative notion of continuity at hand, in order to distinguish degrees of continuity. For this purpose we introduce the notion of a witness to continuity. The following denition of being a witness to continuity is motivated by the role which the scalars α, γplay in the equivalent characterisation of continuity in Remark 2.

Denition 3. Let f, g, h∈ F and β ∈[0,1]. A pair of real numbers (α, γ) ∈ [0,1]² is called a witness-pair to the continuity of % along f, g, h∈ F in β if and only if for allδ∈(α, γ), one has

• δf+ (1−δ)g≺hifβf+ (1−β)g≺hand

• h≺δf+ (1−δ)g ifh≺βf+ (1−β)g, whilst either

• α < β < γ or

• 0 =α=β < γ or

• α < β=γ= 1.

A real numberε∈[0,1]is called a witness to the continuity of%alongf, g, h∈F inβ if and only if there exists someα∈[0,1]orγ∈[0,1]such that either(α, ε) or(γ, ε) is a witness-pair to the continuity of%along f, g, h∈Fin β.

With this denition, we can now rephrase Remark 2:

Remark 4. If%satises completeness (A1a), then %satises continuity (A3) if and only if for all f, g, h∈F and all β ∈[0,1] there exists a witness to the continuity of %along f, g, h∈Finβ.

3 Aggregation of variational preferences

Consider a setN (nite or innite), which we shall call population or electorate.

Elements ofN are called individuals, subsets ofN are called coalitions. Suppose that each individuali∈N is endowed with a variational preference ordering%i

(as dened in Section 2); any such resultingN-sequence%= (%i)_i∈N is called a variational preference prole. In various circumstances for instance, in the course of making certain policy choices the question will arise whether one can aggregate the individual variational preference orderings and obtain a social variational preference ordering (i.e. an aggregate of the individual variational preferences%i which itself happens to be variational preference relation). And if so, are there any rules, satisfying certain rationality conditions, which can be used to assign a (social) variational preference ordering to all variational preference proles or at least to a large class of variational preference proles?

We shall show that any such rule whose domain encompasses a rich class of variational preference proles must be dictatorial in the case of niteNand thus establish an equivalent of Arrow's [1] impossibility theorem for variational preference aggregation. For the case of innite N, we shall rst prove an impossibility result under the assumption of an even more comprehensive aggregator domain, thus obtaining an equivalent of Campbell's [3] impossibility theorem for variational preference aggregation). Secondly, we shall show a possibility result for inniteN under the assumption that the aggregator domain contains only uniformly continuous variational preference prole; this result can be seen as an variational-preference analogue of Fishburn's [10] possibility theorem.

As we shall see in an appendix, on certain restricted domains of proles for nite electorates, the majority voting rule which also satises two important rationality axioms can be used to obtain a social variational preference ordering.

4 Main results: Variational preference aggregation for rich aggregator domains

Denote the set of all variational preference relations onFbyP.

In this paper, a preference aggregator is a mapF with domaindom(F)⊆P^N whose range is a set of complete binary relations onF. A variational preference aggregator or convex risk-preference aggregator is a map F from a subset dom(F)⊆P^N toP. A preference aggregatorF is said to be

• universal if and only ifdom(F) =P^N (so thatF:P^N →P);

• weakly universal if and only ifdom(F)is a rich aggregator domain. Herein, a subsetD⊆P^N is called a rich aggregator domain if and only if there are f, f⁰, g, g⁰∈F and variational preference orderings%1,%2,%3 such that

f %1g, f⁰ %1g⁰, f %2g, f⁰ ≺2g⁰, f ≺3g, f⁰ %3g⁰,and {%1,%2,%3}^N ⊆D;

• systematic if and only if for every%∈dom(F)and allf, f⁰, g, g⁰∈Fwith {i∈N : f %ig}={i∈N : f⁰ %ig⁰} one has

f F

³

%

´

g⇔f⁰F

³

%

´ g⁰;

• Paretian if and only if for every%∈ dom(F) and all f, g∈ F, iff %i g for alli∈N, thenf F

³

%

´ g;

• dictatorial if and only if there exists some i ∈ N (called dictator) such that for every%∈dom(F)and allf, g∈F,

f F³

%´

g⇔f %ig.

The modication weakly in weakly universal is justied:

Remark 5. If S contains at least two elements, then P^N is a rich aggregator domain, and every universal aggregator is also weakly universal.

(All proofs can be found in Appendix C.) Clearly, every dictatorialF can be extended to a universal, systematic and Paretian aggregator. It is remarkable that even the converse holds true:

Theorem 6. LetN be nite and letF be a (variational) preference aggregator.

F is weakly universal, systematic and Paretian if and only if it is dictatorial.

(Theorem 6 is the variational preference analogue of Arrow's [1] possibility theorem.)

Under an additional assumption on the richness of the domain ofdom(F), one can even extend Theorem 6 to the case of inniteN. A prole %is said to be continuous if and only if%iis continuous for alli∈N. Using the terminology of Denition 3, a variational preference prole % is discontinuous in the limit if and only if for all f, g, h∈F and allβ ∈[0,1], everyα∈ [0,1]is a witness to the continuity of %i along f, g, hin β for only nitely manyi ∈N. As an example one might think of a prole of variational preference relations(%i)_i∈N, each with variational representation(u_i, c_i), whereu_i=i u₀ andc_i=c₀ for all i∈N>0.

Theorem 7. Let N be an arbitrary set (nite or innite). Let F be a weakly universal, systematic and Paretian variational preference aggregator.

Suppose that its domaindom(F) contains a prole%that is (continuous, but) discontinuous in the limit. ThenF is dictatorial.

(Theorem 7 can be seen as the variational preference analogue of Campbell's [3] possibility theorem.)

Conversely, one can obtain a possibility result for inniteN by demanding uniform continuity rather than continuity of the variational preference proles in the aggregator domain: A prole(%i)_i∈N is said to be uniformly continuous if and only if for all f, g, h∈F and allβ ∈[0,1], there exist α, γ∈[0,1]which for alli∈N are a witness-pair to the continuity of%i alongf, g, hinβ. Theorem 8. Let N be an innite set, and let D ⊆ P^N be a rich aggregator domain such that all proles in D are uniformly continuous. Then there exist non-dictatorial, weakly universal, systematic and Paretian variational preference aggregatorsF :D→P.

(Theorem 8 is the variational preference analogue of Fishburn's [10]

possibility theorem.)

5 Proof idea

The shortest route in proving the above theorems is to invoke recent results from model aggregation theory, due to Herzberg and Eckert [16] who generalised previous ndings by Lauwers and Van Liedekerke [20]. In order to employ these results, one needs to reformulate the variational preference aggregation problem as a model aggregation problem (see Appendix B); thereafter, the proofs follow relatively easily from the model aggregation theory in Herzberg and Eckert [16]

(see Appendix C). In this section, we briey describe model aggregation theory and its application to the aggregation of variational preferences; a rigorous review can be found in Appendix A.

Model aggregation theory studies the aggregation of rst-order structures (in the sense of mathematical logic). An aggregator in this setting is then just a map from a set of N-sequences of structures of a certain type to a set of structures of such type. It is not dicult to formulate analogues of Arrow's [1]

rationality assumptions in this framework.

Of utmost importance is the notion of a decisive coalition with respect to an aggregatorF: A coalitionD is said to be decisive with respect to an aggregator F if and only if it can be written in the form D = {i∈N : (F,%i)|=φ}

for some prole % ∈ dom(F) and some quantier-free formula φ such that³

F, F³

%´´

|=φ.

Denoting the set of all decisive coalitions with respect toF byDF, one can next prove the following key lemma:

Lemma 9. If F is a weakly universal, systematic and Paretian variational preference aggregator, thenDF is an ultralter on N.²

2An ultralter onNis a nonempty setDof coalitions which is not equal to the powerset of N, is closed under supersets (i.e. ifD ∈ Dand D⁰ ⊇D, then D⁰ ∈ D), closed under intersections (i.e. ifD, D⁰∈D, thenD∩D⁰∈D) and has the property that for any coalition D eitherD∈DorN\D∈D. A lter onN is a set of coalitions that has the rst three properties, but may lack the last one. One can show that ultralters are nothing else but

⊆-maximal lters.

The proof of Lemma 9 uses a slight generalisation of the main lemma in Lauwers and Van Liedekerke [20, Lemma 2]:³ In the proof of that lemma, the ultralter properties (non-triviality; closure under supersets and intersections;

dichotomy) are veried by constructing appropriate proles through exploiting the richness of the aggregator domain.

Since ultralters on nite sets are always principal (i.e. systems of supersets of singletons), Lemma 9 quickly leads to a proof of the only if part in Theorem 6. The proof of the if part in Theorem 6 is straightforward.

Using the ultralter property of the set of decisive coalitions, Theorem 8 and Theorem 7 can now be proved through applications of Šo±'s theorem: For, one can apply Lemma 9 to show that any weakly universal, systematic and Paretian preference aggregator F maps every variational preference prole to the restriction (to the original domain F) of its ultraproduct (with respect to the ultralter DF of decisive coalitions); and conversely, Šo±'s theorem implies that every preference aggregator F which assigns to each variational preference prole indom(F)the restriction of its ultraproduct with respect to a xed ultralter D constitutes a systematic Paretian preference aggregator (which is weakly universal if dom(F) is a rich aggregator domain). Now, since again by Šo±'s theorem restricted ultraproducts preserve universal formulae (also sometimes calledΠ1formulae) that hold in all factor structures, it is clear that the aggregate of a uniformly continuous variational preference prole under a weakly universal systematic Paretian preference aggregator must again be continuous and thus a variational preference prole. Hence, every weakly universal, systematic, Paretian preference aggregator whose domain only consists of uniformly continuous variational preference proles is actually a variational preference aggregator. Now, for inniteN, there exist non-principal ultraltersUonN. Choose such aUand letF :D→Pbe a map whose domain only contains uniformly continuous variational preference proles and which assigns to each element of D the restriction of its ultraproduct with respect to U. This F will then be a variational preference aggregator which is not dictatorial, establishing Theorem 8.

The representation of weakly systematic, Paretian preference aggregators as restricted ultraproduct constructions can also be used to show that no domain of a weakly universal, systematic, Paretian variational preference aggregator can contain a prole % that is discontinuous in the limit. For, if there were such an aggregator, it would on the one hand have to preserve continuity, and on the other hand, every scalar will be a witness to the continuity of only nitely many variational preference orderings in the prole % (which is discontinuous in the limit). A combination of these two facts ultimately implies that the set of decisive coalitions contains a nite set (viz. the set of alli∈N such thatαis a witness to the continuity of%ialongf, g, h∈Finβ, whereinα, β∈[0,1]and f, g, h∈Fhave been chosen such thatαis witness to the continuity ofF

³

%

´

alongf, g, h∈Finβ). But an ultralter which contains a nite set is principal, whence the corresponding aggregator is dictatorial. This proves Theorem 7.

3Whilst the published proof of Lauwers and Van Liedekerke's [20] lemma is incomplete, an addendum by Herzberg, Lauwers, Van Liedekerke and Fianu [17] has recently lled the gap.

6 Extensions

Using the methodology of the present paper, one can also study the aggregation of coherent risk measures for a nite set of states of the world: For, coherent risk measures can be written as negated maxmin expected utility functions, which in turn represent multiple priors preferences, as shown by Gilboa and Schmeidler [13]. Hence, the aggregation of coherent risk measures can be reformulated as an aggregation problem for certainty-independent, continuous, monotonic, uncertainty-averse and non-degenerate weak orders, and the theory of model aggregation can again be used to prove impossibility and possibility results.

Moreover, the approach taken in this paper might perhaps also be used to analyse the aggregation of dynamic variational preferences and thus of dynamic convex risk measures: For, the representation theorem of Föllmer and Schied [11, 12] has been extended to a dynamic setting by Detlefsen and Scandolo in a paper on dynamic convex risk measures [4] which builds upon on Riedel's seminal article on dynamic coherent risk measures [27]. Moreover, Maccheroni, Marinacci and Rustichini [24] have recently developed a dynamic generalisation of their axiomatisation of variational preferences [23]. Combining their theorem with Detlefsen and Scandolo's result, one obtains a decision-theoretic foundation of dynamic convex risk measures in terms of dynamic variational preferences.

At a more technical frontier, the systematicity condition can possibly be relaxed, since systematicity is equivalent to the weaker aggregator condition of independence if the conditional entailment relation among the set of test sentences has full transitive closure.

7 Conclusion

We have formulated Arrow-type aggregation problems for convex risk measures or variational preferences. Choosing a methodology inspired by Lauwers and Van Liedekerke [20], one can prove analogues of Arrow's impossibility theorem, Campbell's impossibility theorem, and Fishburn's possibility theorem. The proof method is suciently general to be applied to Arrow-type aggregation of coherent risk measures or multiple priors preferences, and perhaps even dynamic convex or dynamic coherent risk measures and their variational counterparts.

Appendices

A Review of model aggregation theory

The theory of model aggregation was begun by Lauwers and Van Liedekerke [20] (see also Herzberg, Lauwers Van Liedekerke and Fianu [17]) and continued recently by Herzberg and Eckert [15, 16]. In the following, we only review special cases of the most important known results from model aggregation with particular relevance for the analysis of variational preference aggregation. In fact, the presentation in this appendix is only slightly more general than the work of Lauwers and Van Liedekerke [20] in that it allows for arbitrarily many predicate symbols rather than just one , whence any reader who knows Lauwers and Van Liedekerke's work [20] may well skip this appendix. For proofs

and a more general account of model aggregation, see Herzberg and Eckert [15, 16].

We assume in this section that the reader has some basic knowledge of model theory. The paper by Lauwers and Van Liedekerke [20] contains a short introduction to logic and model theory for social choice theorists; more comprehensive introductions can be found in textbooks such as those by Bell and Slomson [2] or Hodges [18].

Let A be a set. Let L be a language consisting of predicate symbols P˙n, n∈κ, and constant symbolsa˙ for all elements aofA. The arity of P˙n will be denoted δ(n), for alln∈κ.

For the purposes of this paper, anL-structure is a pairB= D

B,­ P_n^B®

n∈κ

E

consisting of a setB⊇A(called the domain ofB) and certain setsP_n^B ⊆B^δ(n) which serve to interpret the predicate symbols P˙n through Tarski's denition of truth. We require that by denition anyL-structure interprets the constant symbolsa˙ canonically, i.e. bya, for anya∈A.

LetS be the set of atomic formulae in L. LetT be the Boolean closure of S, i.e. the closure ofSunder the logical connectives¬˙,∧˙,∨˙. The elements ofT are called test sentences, and the elements ofS are called basic test sentences.

Let T be a consistent set of universal sentences in L.⁴, and let Ω be the collection of models of T with domain A. As is usual in model theory, the restriction of anL-structureBis theL-structure that is obtained by restricting the interpretations of the relation symbol to the domainA; it is denotedresAB.

We assume that there are two sentences inS, henceforth denoted µ, ν ∈S, such that each ofµ∧ν,˙ µ∧˙¬ν˙ and¬µ˙ ∧ν˙ is consistent withT, in symbols,

T∪ {µ∧ν˙ } 6` ⊥, T∪ {µ∧˙¬ν˙ } 6` ⊥, T∪ {¬µ˙ ∧ν} 6` ⊥˙ (1) (wherein ⊥is shorthand forφ∧˙¬φ˙ for some sentenceφ).

Since S is the set of all atomic formulae in L and T is a set of universal sentences, the following propositions hold for allL-structuresAand allA1,A2∈ Ω:

(∀λ∈S (A1|=λ⇔A2|=λ))⇒A1=A2. (2) A|=T ⇒resAA∈Ω (3)

∀λ∈T (A|=λ⇔resAA|=λ). (4) Elements of Ω^N will be called proles. An aggregator is a map f whose domaindom(f)is a subset ofΩ^N and whose range is a subset of Ω.⁵

For allλ ∈ T and all ω ∈ Ω^N, we denote the coalition supporting λ given proleω, by

C(ω, λ) :={i∈N : ωi|=λ}. Let us x an aggregatorf. Consider the following axioms:

4A sentence is universal if it (in its prenex normal form) has the form ˙( ˙∀v˙k₁˙)· · ·˙( ˙∀v˙k_m˙)φ for some formulaφthat does not contain any quantiers and some nonnegative integerm.

5We deviate from Lauwers' and Van Liedekerke's [20] notation as follows:

• Aggregators will be denoted byf (instead ofAF).

• Proles will be denoted byωorhωi : i∈Ni(instead ofhAi : i∈Ni).

• The image of a proleωunder an aggregator f will be denoted byf(ω)(instead of A(ω)).

(A1). dom(f) = Ω^N.

(A1'). There exist modelsA1,A2,A3∈Ωsuch that 1. A1|=µ∧ν˙ ,A2|=µ∧˙¬ν,˙ A3|= ˙¬µ∧ν˙ , and 2. {A1,A2,A3}^N ⊆dom(f).

(A2). For allω∈dom(f)and allλ∈T, iff(ω)|=λ, thenC(ω, λ)6=∅.

(A3). For allω, ω⁰ ∈dom(f)and allλ, λ⁰ ∈Tsuch thatC(ω, λ) =C(ω⁰, λ⁰), one hasf(ω)|=λif and only iff(ω⁰)|=λ⁰.

(A1) is the axiom of Universality. Axiom (A2) is a generalised Pareto Principle. (A3) is a generalised form of the axiom of Systematicity, which itself is a strong variant of the axiom of Independence of Irrelevant Alternatives.⁶

By our assumptions onµ, ν∈S, there must beL-structuresA1,A2,A3such that A1,A2,A3|=T as well asA1|=µ∧ν˙ ,A2|=µ∧˙¬ν˙ ,A3|= ˙¬µ∧ν˙ . SinceT is universal and so are all elements ofT, we may assume thatA1,A2,A3 all have domainA(otherwise, take their restriction toA). Hence, Axiom (A1') is simply a weak version of (A1) because of our assumption about the sentencesµ, ν∈S.

Given an aggregatorf, we dene the set of decisive coalitions by Df :={C(ω, λ) : ω∈dom(f), λ∈T, f(ω)|=λ}.

It is not dicult to verify that systematic aggregators are characterised by their sets of decisive coalitions:

Remark 10. If f satises (A3), then for allω∈dom(f) andλ∈T, C(ω, λ)∈Df ⇔f(ω)|=λ.

This framework is suciently general to cover the cases of preference aggregation, propositional judgment aggregation, and modal aggregation.⁷ The general model aggregation theory in Herzberg and Eckert [15, 16] admits more general sets of test sentences T and relaxes the aggregator axioms (A2) and (A3).

The key result of model aggregation is the following lemma:⁸

6Systematicity vacuously implies Independence of Irrelevant Alternatives. The converse is true under additional hypotheses: In the preference aggregation framework, the combination of Independence of Irrelevant Alternatives and the Pareto Principle implies Systematicity if the individual preferences are complete and quasi-transitive (cf. Lauwers and Van Liedekerke [20, Section 6, p. 232]).

7For example, for preference aggregation, one lets L have a single binary predicate P˙, modelling the preference relation. The setAwill be the set of alternatives. The interpretation of P˙( ˙a,b)˙ will be ais preferred to b. (Thus, the interpretation of ωi |= ˙P( ˙a,b)˙ is under prole ω, individual i prefers ato b, and the interpretation of f(ω) |= ˙P( ˙a,b)˙ is under proleω,ais socially preferred tob.) T can be any universal theory in that language. For propositional judgment aggregation, one letsLhave a single unary predicateB˙, modelling a belief operator. The setAwill be the agenda. The interpretation ofB˙a˙ais accepted. (Thus, the interpretation ofωi|= ˙Ba˙is under proleω, individualiacceptsa, and the interpretation off(ω)|= ˙Ba˙is under proleω,ais socially accepted.) T can be any universal theory in that language.

8Lemma 11 slightly generalises the main lemma in Lauwers and Van Liedekerke [20, Lemma 2]; a proof in a more general setting can be found in Herzberg and Eckert's rst paper [15].

Lemma 11. Let f be weakly universal, systematic, and Paretian. Then,Df is an ultralter.⁹

We say thatf is dictatorial if and only if there exists someif ∈N (called the dictator) such thatDf ={J ⊆N :if ∈J}.

Remark 12. Let f be an aggregator, and suppose N is nite. Then, f is dictatorial if and only if Df is an ultralter.

As a corollary of the ultralter property of the set of decisive coalitions (see Lemma 11), we then get an analogue of Arrow's impossibility theorem:¹⁰ Corollary 13 (Impossibility theorem). Let f be weakly universal, systematic, and Paretian. If N is nite, thenf is dictatorial.

By Šo±'s theorem [22]:

Remark 14. If Dis an ultralter, then resA

Y

i∈N

ωi/D|=λ⇔C(ω, λ)∈D for allω∈Ω^N andλ∈T.

Lemma 15. Letf be weakly universal, systematic, and Paretian, thenf(ω) = resA

Q

i∈Nωi/Df for all ω∈dom(f).

Lemma 16. SupposeDis an ultralter, and consider the aggregatorresA

Q/D, dened by

resA

Y/D: Ω^N →Ω, ω7→resA

Y

i∈N

ωi/D.

ThenresA

Q/D is a universal, systematic and Paretian aggregator.

Let βN denote the set of all ultralters on the set N, and let AR be the set of all universal, systematic and Paretian aggregators. Now one can prove a general version of the KirmanSondermann [19] correspondence:¹¹

Theorem 17 (KirmanSondermann correspondence). There is a bijectionΛ :AR→βN,given by

∀f ∈AR Λ(f) =Df. Its inverse is given by

∀D∈βN Λ⁻¹(D) = resA

Y/D,

wherein, as in Lemma 16,resA

Q/D:ω7→resA

Q

i∈Nωi/D.

9For the denition of an ultralter, see footnote 2 on page 7.

10

11This Theorem 17 is a slight generalisation of Lauwers and Van Liedekerke's main theorem;

its proof in a more general framework than that of the present paper can be found in Herzberg and Eckert [15].

Consider an arbitraryL-sentence which is not universal. In its prenex normal form it can be written as ψ≡ ˙( ˙∀x˙1˙). . .˙( ˙∀x˙m˙)˙( ˙∃y˙˙)φ( ˙x1, . . . ,x˙m; ˙y), wherein m is a nonnegative integer andφ( ˙x1, . . . ,x˙m; ˙y) is an L-formula with m+ 1free variables. For the rest of this section,ψandφare xed in this manner.

We say that a proleω∈Ω^N has nite witness multiplicity with respect to ψif and only ifωi|=ψfor alli∈N, but for alla1, . . . , am, a⁰∈A, the coalition {i∈N : ω|=φ(a1, . . . , am;a⁰)}is nite.

An aggregator f is said to preserve an L-sentence ψ if and only if for all ω ∈dom(f), one has f(ω)|=ψ wheneverωi |=ψ for alli∈N. We then have the following theorem:¹²

Theorem 18. Let f be weakly universal, systematic and Paretian, suppose f preserves ψ, and assume that there exists some ω ∈ Ω^I with nite witness multiplicity with respect to ψ. Then,f is a dictatorship.

B Variational preference aggregation as model aggregation

As we have remarked before, our proofs depend largely on the recent results on model aggregation by Herzberg and Eckert [15, 16] that generalise previous work by Lauwers and Van Liedekerke [20]. The key to the proofs of Theorem 6 and Theorem 7 is therefore the rephrasing of the variational preference aggregation problem in the framework of rst-order model theory.

The formulation of the variational preference aggregation problem in the framework of rst-order model theory can even be accomplished without appealing to multi-sorted predicate logic, as it will turn out that one can identify the closed unit interval [0,1] ⊆ R and the open unit interval (0,1) ⊆ R with subsets ofFc and hence ofF. The domain of the model-theoretic structures to be aggregated will thus be justF, and individual constant and variable symbols will always be interpreted as referring to constant or variable elements ofF.

In order to embed the closed and open unit intervals ofRintoF, choose two distinct elements x0, x1∈X, and dene for allα∈[0,1]a constant functionαˇ by

ˇ

α:s7→αx0+ (1−α)x1. Clearly, the mapα7→αˇ is injective.¹³ Hence, if we dene

I¯={αˇ : α∈[0,1]}

and

I={αˇ : α∈(0,1)}= ¯I\ {x0, x1},

12This theorem is in some sense an abstract version of a similar result by Lauwers and Van Liedekerke [20, p. 230, Property 4]; its proof can be found in Herzberg and Eckert's second paper [16].

13For, if

αx0+ (1−α)x1=βx0+ (1−β)x1

for someα, β∈[0,1]withα6=β, then

(α−β)x0= (α−β)x1

and thusx0=x1, contradiction.

there is a canonical bijection betweenI¯and[0,1]⊆Ras well as betweenI and (0,1)⊆R.

This allows us to dene a mixture operatorm: ¯I×F² →F as follows: For allα∈[0,1]andf, g∈F, put

m(ˇα;f, g) =αf+ (1−α)g∈F.

(Recall that Fis a convex subset of the vector spaceY^S.) For everys∈S, let πs:F→Fc be the projection operator which mapsf to the constant function with range{f(s)}, so thatπs(f) :s7→f(s)for allf ∈F.

Finally, one can dene a linear ordering <I¯ onI¯by ˇ

α <I¯βˇ⇔α < β for allα, β∈[0,1].

With these denitions, we may now consider the following axioms for a binary relation %with symmetric part ∼ (i.e. f ∼g if and only if f %g and g%f) and asymmetric partÂ(i.e. f Âg if and only iff %g butg6∼f):

(A1) Weak order properties. For all f, g ∈ F, either f % g or g % f; for all f, g, h∈F, iff %g andg%h, thenf %h.

(A2) Weak certainty independence. For allf, g∈F,x, y∈Fc anda∈I, if m(a;f, x)%m(a;g, x),

then

m(a;f, y)%m(a;g, y).

(A3) Continuity. For allf, g, h∈Fand allb∈I, there exista, c∈I¯such that if m(b;f, g) % h, then for all d ∈ I with a <I¯d <I¯c, one has

m(d;f, g)%h, and

if h % m(b;f, g), then for all d ∈ I with a <I¯d <I¯c, one has h % m(d;f, g),

while either

x0≤I¯a <I¯b <I¯c≤I¯x1 or x0=a=b <I¯c≤I¯x1or x0≤I¯a < b=c=x1.

(A4) Monotonicity. For allf, g∈F, ifV

s∈Sπs(f)%πs(g), thenf %g.

(A5) Uncertainty aversion. For all f, g ∈ F and a ∈ I, if f ∼ g, then m(a;f, g)%f.

(A6) Non-degeneracy. There existf, g∈Fsuch thatf Âg.

All these axioms can be captured in a language of rst-order logic with:

• two unary predicate symbolsC,˙ I˙ (expressing membership in the subsets Fc andI, respectively, ofF),

• two binary predicate symbols%˙ and <˙I¯,

• card(S)operator symbols π˙_s,

• and a ternary operation symbolm.˙

Henceforth, the language with the predicate symbolsC,˙ I˙, the operator symbols

˙

πs(for eachs∈S), the predicate symbols%˙ and <˙I¯ the operation symbolm˙, and a constant symbolf˙for every elementf ∈F.

LetΓ be the set of all modelsA= D

F, D

C^A, I^A,%^A, <I¯A,­ π_s^A®

s∈S, m^A EE

of (A1-A6) with domainFsuch thatAcanonically interprets

• the constant symbolsf˙(i.e. f^A=f for every f ∈F),

• the unary predicate symbolsC˙ andI˙(i.e. C^A=Fc andI^A=I),

• the binary relation symbol <˙I¯ (i.e. x˙^A<I¯Ay˙^Aif and only ifx <I¯y for all x, y∈X),

• the operator symbolsπ˙s(i.e. A|= ˙πs( ˙f) ˙= ˙g if and only iff(s) =g for all s∈S andf, g∈F), and

• the ternary operation symbolm (i.e. A |= ˙m³

˙ a; ˙f ,g˙´

˙

= ˙hif and only if m(a;f, g) =hfor allf, g, h∈Fand alla∈I).

Then, elements of Γ are in a canonical one-to-one correspondence with variational preference orderings. Hence, variational preference aggregators are in a canonical one-to-one correspondence with maps G : dom(G) → Γ where dom(G)⊆Γ^N; such mapsg shall also be called model aggregators.

One can now rephrase the variational preference aggregator axioms as model aggregator axioms. LetTbe the Boolean closure of the set of atomic sentences.

We shall callT the set of test sentences.

• universal if and only ifdom(G) = Γ^N (so thatG: Γ^N →P);

• weakly universal if and only ifdom(G)is a rich aggregator domain. Herein, a set D is a rich aggregator domain if and only if there exist atomic sentencesµ, ν and modelsA1,A2,A3 such that

A1|=µ∧ν,˙ A2|=µ∧˙¬ν,˙ A2|= ˙¬µ∧ν˙ and {A1,A2,A3}^N ⊆dom(G);

• systematic if and only if for everyA∈dom(G)and all test sentencesλ, λ⁰ satisfying{i∈N : Ai|=λ}={i∈N : Ai|=λ⁰}one has

G(A)|=λ⇔G(A)|=λ⁰;

• Paretian if and only if for everyA∈dom(G)and all test sentencesλ, if AI |=λfor alli∈N, thenG(A)|=λ;

• dictatorial if and only if there exists some i ∈ N (called dictator) such that for everyA∈dom(G)and all test sentencesλ,

G(A)|=λ⇔Ai |=λ.

A coalitionD is said to be decisive with respect to a model aggregatorGif and only if there is someA∈dom(G)and some test sentenceλsuch that

G(A)|=λ, D={i∈N : Ai|=λ}. The set of all decisive coalitions with respect toGis denotedDG.

C Proof details

Let∆be the set of all probability measures onS. By the MacceroniMarinacci Rustichini theorem [23, Theorem 3], a binary relation %on Fis a variational preference relation if and only if there exists a nonzero linear functionu:X →R and a convex lower-semicontinuous functionc: ∆→[0,+∞]whose inmum is a real number (rather than−∞or +∞) such that for all f, g∈F,

f %g⇔min

p∈∆

µZ

u◦f dp+c(p)

¶

≥min

p∈∆

µZ

u◦gdp+c(p)

¶ . In that case, we say that%has the variational representation(u, c).

Proof of Remark 5. Let s0, s1 be two distinct elements. Let u be a nonzero linear function. Without loss of generality, assumeu(x0)< u(x1). Letc1, c2, c3

be such thatci(p) = +∞for alli∈ {1,2,3} and allp∈∆ such that p{s}>0 for somes∈S\{s0, s1}. Then, for everyi∈ {1,2,3}, there exists some function c⁰_i: [0,1]→Rsuch that

c⁰_i(q) =ci(qδs0+ (1−q)δs1)

for allq∈[0,1](whereinδsdenotes the Dirac probability measure concentrated on the singleton{s}). Letf, g∈Fbe such that

f(s0) =x0, f(s1) =x1, g(s0) =x1, g(s1) =x0. Clearly then for anyi∈ {1,2,3}, one has

minp∈∆

µZ

u◦f dp+ci(p)

¶

= min

q∈[0,1](qu(x0) + (1−q)u(x1) +c⁰_i(q))

= u(x1) + min

q∈[0,1](q(u(x0)−u(x1)) +c⁰_i(q)) minp∈∆

µZ

u◦gdp+ci(p)

¶

= min

q∈[0,1](qu(x1) + (1−q)u(x0) +c⁰_i(q))

= u(x0) + min

q∈[0,1](q(u(x1)−u(x0)) +c⁰_i(q)) Let us now put c⁰₁(q) = 0 for all q ∈ [0,1]. Then, because u(x1) > u(x0)or equivalentlyu(x0)−u(x1)<0 andu(x1)−u(x0)>0, we have

minp∈∆

µZ

u◦f dp+c1(p)

¶

= u(x1) + min

q∈[0,1]



q(u(x0)−u(x1)) +c⁰₁(q)

| {z }

=0





= u(x1) +u(x0)−u(x1) =u(x0) minp∈∆

µZ

u◦gdp+c1(p)

¶

= u(x0) + min

q∈[0,1]



q(u(x1)−u(x0)) +c⁰₁(q)

| {z }

=0





= u(x0)

hence f ∼1 g if %1 is chosen as the variational preference relation with variational representation(u, c1).

Next, putc⁰₂:q7→q(u(x1)−u(x0)). Then, minp∈∆

µZ

u◦f dp+c2(p)

¶

= u(x1) + min

q∈[0,1](q(u(x0)−u(x1)) +c⁰₂(q))

= u(x1) minp∈∆

µZ

u◦gdp+c2(p)

¶

= u(x0) + min

q∈[0,1](q(u(x1)−u(x0)) +c⁰₂(q))

= u(x0) + min

q∈[0,1]2q(u(x1)−u(x0))

= u(x0),

hence f Â2 g if %2 is chosen as the variational preference relation with variational representation(u, c2).

Finally, putc⁰₃:q7→q(u(x0)−u(x1)). Then, minp∈∆

µZ

u◦f dp+c3(p)

¶

= u(x1) + min

q∈[0,1](q(u(x0)−u(x1)) +c⁰₃(q))

= u(x1) + min

q∈[0,1]2q(u(x0)−u(x1))

= u(x1) + 2 (u(x0)−u(x1))

= 2u(x₀)−u(x₁)< u(x₀) minp∈∆

µZ

u◦gdp+c3(p)

¶

= u(x0) + min

q∈[0,1](q(u(x1)−u(x0)) +c⁰₃(q))

= u(x₀)

hence g Â3 f if %3 is chosen as the variational preference relation with variational representation(u, c3).

All in all, we have found variational preference relations%1,%2,%3 with f %1g%1f, f Â2g, gÂ3f.

If we put f⁰ = g and g⁰ = f, then f, g, f⁰, g⁰ and %1,%2,%3 satisfy the requirements in the denition of a rich aggregator domain.

It follows thatP^N is a rich aggregator domain. Therefore, every universal aggregator is also weakly universal.

Proof of Theorem 6. The reformulation of variational preference aggregation as model aggregation in Appendix B permits the application of the impossibility result in Corollary 13 (a generalisation of Arrow's theorem) which in our context says that any weakly universal, systematic, Paretian model aggregator which preserves the (universal) axioms A1-A2, A4-A6 (i.e. the variational preference axioms without continuity) is a dictatorship ifN is nite. Hence, a fortiori, any variational preference aggregator (which by denition even preserves all axioms A1-A6 on its domain) must be a dictatorship if N is nite.

Proof of Theorem 7. The reformulation of variational preference aggregation in Appendix B also allows us to use the characterisation of model aggregators as restricted ultraproduct constructions (Lemma 15). In order to apply

Theorem 18, the impossibility result for aggregators on innite populations which preserve certain non-universal formulae (for a similar result, cf. Lauwers and Van Liedekerke [20, p. 230, Property 4]), observe rst that continuity of variational preferences is not a universal formula and secondly that any prole which is discontinuous in the limit has nite witness multiplicity with respect to continuity (in the terminology of Theorem 18, the abstract version of a similar).

Proof of Theorem 8. Again, in light of Appendix B, we may use the characterisation of aggregators as restricted ultraproduct constructions (Lemma 15). Note that for xed f, g, h and α, β, γ, the formula hα, γi is a witness-pair to the continuity of % along f, g, h in β is a universal formula, and all the axioms A1-A2, A4-A6 are also universal formulae. Hence, the axioms A1-A2, A4-A6 as well as the formulae hα, γi is a witness-pair to the continuity of % along f, g, h in β (for all xed f, g, h, α, β, γ) are preserved by restricted ultraproducts. Therefore, restricted ultraproduct constructions on rich domains are model aggregators which not only are weakly universal, systematic and Paretian and preserve axioms A1-A2, A4-A6, but they also aggregate uniformly continuous proles into continuous proles. Hence, any restricted ultraproduct construction on a rich domain with only uniformly continuous proles constitutes weakly universal, systematic and Paretian variational preference aggregator. However, on an innite setN there are non- principal ultralters and thus non-dictatorial aggregators derived from restricted ultraproduct constructions.

Proof of Lemma 9. This is a direct consequence of Lemma 11, itself a slight generalisation of the main lemma in Lauwers and Van Liedekerke [20, Lemma 2].

D Variational preference aggregation with restricted domain through majority voting

We have seen that in general, universal systematic Paretian aggregation of convex risk measures is impossible. If one drops universality, then rational aggregation of risk measures is still possible, viz. through majority voting about risk measures, but at the expense of considerable restrictions on the variational preference proles.

In analysing majority decisions about convex risk measures, one can build on the work of Dietrich and List [7] who have developed a theory of majority voting in the very general framework of judgment aggregation, including a generalisation of May's [25] theorem (which uniquely characterises majority voting by means of certain axioms such as anonymity and acceptance/rejection neutrality). In order to do so, one has to embed the aggregation problem for risk measures into the framework of judgment aggregation.

Consider the axiom systemΣconsisting of the following formulae:

• The axioms (A1-A6) as reformulated in Appendix B.

• The formulae¬˙f˙= ˙˙g for allf, g∈Fsuch thatf 6=g.

• All formulae of the formπ˙s( ˙f) ˙= ˙xfor alls∈S,f ∈Fandx∈Fcsatisfying πs(f) =x.

• All formulae of the formC˙x˙ for all x∈Fc, and all formulae of the form

˙

¬C˙f˙for allf ∈F\Fc.

• All formulae of the formI˙a˙ for alla∈I, and all formulae of the form¬˙I˙f˙ for allf ∈F\I.

• All formulae of the form m˙ ³

˙ a; ˙f ,g˙´

= ˙h for all a ∈ I and f, g, h ∈ F satisfyingm(a;f, g) =h.

For any set of L-formulaeA, completeness and consistency will be understood to meanΣ-completeness andΣ-consistency

Let

Ξ =n

f˙%˙g,˙ ¬˙f˙%˙g˙ : f, g∈Fo .

This is an agenda in the terminology of judgment aggregation, i.e. a set of proposition-negation pairs. In the following, for anyp∈Ξ, we mean by¬p¯ the other element of the proposition-negation pair inΞto whichpbelongs, so that

¯

¬¯¬p=pfor anyp∈Ξ.

A fully rational judgment set is a complete and consistent subset ofΞ; note that by the choice of Σ, any fully rational judgment set uniquely determines a preference relation % on F that satises axioms (A1-A6). The set of fully rational judgment sets will be denoted byD.

Let N be nite. A prole is an N-tuple A = (Ai)_i∈N of fully rational judgment sets. For each prole A and any p ∈ Ξ, we dene the coalition supporting punder prole Aby

A(p) ={i∈N : p∈Ai}.

The aggregation rule of majority voting is then dened as the map F:D^N →2^Ξ, A7→ {p∈Ξ : card (A(p))>card (A(¯¬p))}. At least whenever card(N) is odd, the aggregate judgment set F(A) will be complete for every A∈D^N. The question is whether F(A) will be consistent as well; if it is, it is a fully rational judgment set and thus by our observation made above, uniquely determines a preference relation % on F that satises axioms (A1-A6), hence a convex risk measure.

A sucient condition for the consistency ofF(A) for certain A∈D^N has been discovered by Dietrich and List [7] and is known as the value-restriction property. A proleA∈D^N value-restricted is for every non-singleton, minimal inconsistent subset Y ⊆Ξ there exists a two-element subsetZ ⊆Y such that Z 6⊆Aifor alli∈N. If there is an ordering on the agenda with respect to which every Ai (for i ∈N) is (locally) single-plateaued or single-canyoned, then the prole is value-restricted and henceF(A)is consistent (and complete anyway).

However, in the case of variational preference proles, the conditions of single-canyonedness or single-plateauedness let alone the value-restriction property do not appear to be natural conditions. This gives additional weight to the impossibility results in the present paper.